Sure, I'd be happy to help explain these problems and solutions in detail!
### Question 7:
Find the angle whose supplement is four times its complement.
To find the angle [tex]\(x\)[/tex] degrees:
1. Define the Complement and Supplement:
- The supplement of an angle [tex]\(x\)[/tex] is [tex]\(180^{\circ} - x\)[/tex].
- The complement of an angle [tex]\(x\)[/tex] is [tex]\(90^{\circ} - x\)[/tex].
2. Set Up the Equation:
According to the problem, the supplement of the angle is four times its complement:
[tex]\[
180^{\circ} - x = 4 \times (90^{\circ} - x)
\][/tex]
3. Solve the Equation:
Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
180 - x = 4 \times (90 - x)
\][/tex]
[tex]\[
180 - x = 360 - 4x
\][/tex]
[tex]\[
3x = 180
\][/tex]
[tex]\[
x = 60
\][/tex]
So, the angle whose supplement is four times its complement is [tex]\(60^{\circ}\)[/tex].
### Question 8:
Find the measure of an angle, If seven times its complement is [tex]$10^{\circ}$[/tex] less than three times its supplement.
To find the angle [tex]\(y\)[/tex] degrees:
1. Define the Complement and Supplement:
- The complement of an angle [tex]\(y\)[/tex] is [tex]\(90^{\circ} - y\)[/tex].
- The supplement of an angle [tex]\(y\)[/tex] is [tex]\(180^{\circ} - y\)[/tex].
2. Set Up the Equation:
According to the problem, seven times the complement of the angle is [tex]\(10^{\circ}\)[/tex] less than three times the supplement of the angle:
[tex]\[
7 \times (90^{\circ} - y) = 3 \times (180^{\circ} - y) - 10^{\circ}
\][/tex]
3. Solve the Equation:
Simplify and solve for [tex]\(y\)[/tex]:
[tex]\[
7 \times (90 - y) = 3 \times (180 - y) - 10
\][/tex]
[tex]\[
630 - 7y = 540 - 3y - 10
\][/tex]
[tex]\[
630 - 7y = 530 - 3y
\][/tex]
[tex]\[
4y = 100
\][/tex]
[tex]\[
y = 25
\][/tex]
So, the angle such that seven times its complement is [tex]\(10^{\circ}\)[/tex] less than three times its supplement is [tex]\(25^{\circ}\)[/tex].
Thus:
- The angle in Question 7 is [tex]\(60^{\circ}\)[/tex].
- The angle in Question 8 is [tex]\(25^{\circ}\)[/tex].
